Question

Verify all vector space axioms and determine, if the set of all triples of real numbers (x, y, z) is a vector space under the given operations (x, y, z) + (x', y', z') = (x + x', y + y', z + z') and k(x, y, z) = (kx, y, z)

Comments

Can you define what you mean by x' in relation to x? (Similarly for y' and z'.)

The distributive rule is usually k(x,y,z)=(kx,ky,kz), where k is a real scalar. Did you mean to state k(x,y,z)=(kx,y,z)?

If so, the vector space axiom for distribution doesn't apply.

Answer 1

The distributive rule is one of the axioms, so k(x,y,z)=(kz,ky,kz) under that axiom.

If the operation k(x,y,z)=(kz,y,z) then the proposition, based on the given operations, that it fulfils the required axioms is false. All axioms must be satisfied.